Simulations: Table 1, 2, 4 and Figure 2.

n = 60

##     alpha_1 alpha_n/2+1      beta_1  beta_n/2+1      gamma1 
##   0.1570392   0.2112906   0.1925428   0.1753245   0.0136311
##             t=0.4 t=0.6 6=0.8 t=0.4 t=0.6 6=0.8
## alpha_1     0.953 0.937 0.940  1.06  1.61  1.50
## alpha_n/2+1 0.940 0.938 0.930  1.62  1.71  1.44
## beta_1      0.927 0.942 0.940  1.17  1.63  1.36
## beta_n/2+1  0.943 0.956 0.935  1.22  1.65  1.68
## gamma1      0.921 0.918 0.903  0.36  0.48  0.36

n = 100

##     alpha_1 alpha_n/2+1      beta_1  beta_n/2+1      gamma1 
## 0.133081737 0.185719703 0.125437910 0.171795708 0.007900022
##             t=0.4 t=0.6 6=0.8 t=0.4 t=0.6 6=0.8
## alpha_1     0.944 0.929 0.929  1.00  1.58  1.44
## alpha_n/2+1 0.940 0.927 0.936  1.63  1.67  1.38
## beta_1      0.927 0.940 0.948  1.12  1.62  1.29
## beta_n/2+1  0.945 0.930 0.953  1.18  1.62  1.66
## gamma1      0.921 0.932 0.916  0.29  0.40  0.30

n = 200

##     alpha_1 alpha_n/2+1      beta_1  beta_n/2+1      gamma1 
##  0.11425551  0.17558398  0.10656619  0.16164565  0.00417785
##             t=0.4 t=0.6 6=0.8 t=0.4 t=0.6 6=0.8
## alpha_1     0.946 0.934 0.940  0.95  1.55  1.39
## alpha_n/2+1 0.943 0.957 0.948  1.64  1.66  1.30
## beta_1      0.910 0.942 0.948  1.08  1.59  1.22
## beta_n/2+1  0.949 0.945 0.932  1.16  1.61  1.63
## gamma1      0.926 0.933 0.925  0.22  0.31  0.23

n = 500

##     alpha_1 alpha_n/2+1      beta_1  beta_n/2+1      gamma1 
## 0.099349014 0.173573538 0.099535479 0.164397190 0.002378747
##             t=0.4 t=0.6 6=0.8 t=0.4 t=0.6 6=0.8
## alpha_1     0.958 0.950 0.934  0.92  1.55  1.33
## alpha_n/2+1 0.948 0.942 0.949  1.69  1.65  1.23
## beta_1      0.919 0.927 0.956  1.07  1.62  1.14
## beta_n/2+1  0.947 0.954 0.953  1.15  1.58  1.59
## gamma1      0.934 0.921 0.932  0.15  0.22  0.16

Simulations: Table 3: Different initial values

Ini = 0; True

##     alpha_1 alpha_n/2+1      beta_1  beta_n/2+1      gamma1 
## 0.133077817 0.185713300 0.125434164 0.171789684 0.007899757
##             t=0.4 t=0.6 6=0.8 t=0.4 t=0.6 6=0.8
## alpha_1     0.944 0.929 0.929  1.00  1.58  1.44
## alpha_n/2+1 0.940 0.927 0.936  1.63  1.67  1.38
## beta_1      0.927 0.940 0.948  1.12  1.62  1.29
## beta_n/2+1  0.945 0.930 0.953  1.18  1.62  1.66
## gamma1      0.921 0.932 0.916  0.29  0.40  0.30
##   X        x
## 1 1 1.039422

Ini = -1

##     alpha_1 alpha_n/2+1      beta_1  beta_n/2+1      gamma1 
## 0.129768611 0.179779581 0.128752232 0.168349320 0.007548707
##             t=0.4 t=0.6 6=0.8 t=0.4 t=0.6 6=0.8
## alpha_1     0.944 0.936 0.944  1.00  1.59  1.44
## alpha_n/2+1 0.945 0.947 0.956  1.65  1.68  1.37
## beta_1      0.916 0.936 0.938  1.11  1.62  1.29
## beta_n/2+1  0.947 0.949 0.948  1.18  1.63  1.66
## gamma1      0.930 0.923 0.911  0.29  0.40  0.29
##   X       x
## 1 1 1.06203

Ini = 2

##     alpha_1 alpha_n/2+1      beta_1  beta_n/2+1      gamma1 
##  0.13092288  0.17858651  0.12892922  0.17152887  0.00769082
##             t=0.4 t=0.6 6=0.8 t=0.4 t=0.6 6=0.8
## alpha_1     0.957 0.939 0.936  1.00  1.58  1.45
## alpha_n/2+1 0.946 0.940 0.929  1.63  1.68  1.38
## beta_1      0.904 0.953 0.930  1.11  1.62  1.30
## beta_n/2+1  0.948 0.951 0.941  1.19  1.63  1.66
## gamma1      0.940 0.938 0.921  0.29  0.40  0.30
##   X        x
## 1 1 1.870246

Ini = 5

##     alpha_1 alpha_n/2+1      beta_1  beta_n/2+1      gamma1 
## 0.127662068 0.180452260 0.131380474 0.164902633 0.007771576
##             t=0.4 t=0.6 6=0.8 t=0.4 t=0.6 6=0.8
## alpha_1     0.948 0.930 0.932  1.00  1.57  1.44
## alpha_n/2+1 0.949 0.934 0.941  1.64  1.67  1.37
## beta_1      0.915 0.939 0.939  1.11  1.62  1.29
## beta_n/2+1  0.947 0.958 0.953  1.19  1.62  1.65
## gamma1      0.920 0.919 0.912  0.29  0.40  0.30
##   X        x
## 1 1 3.771921

Simulations: Figure 3: Bias of K’th method

Simulations: Z dependent

rho = 0.2

rho = 0.4

rho = 0.6

rho = 0.8

Simulation: Goodness of fit Case 1

Our’s method ( effect)

K’s method (homo effect)

Simulation: Goodness of fit Case 2

Our’s method ( effect)

K’s method (homo effect)

Simulation: Goodness of fit Case 3

Our’s method ( effect)

K’s method (homo effect)

Simulations: Cross Validation.

MIT Analysis

Gamma estimation

Gof: Our’s method ( effect)

Gof: K’s method (homo effect)

Bike Analysis

Alpha Beta estimation

## Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
## ℹ Please use `linewidth` instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.
## Warning: The `size` argument of `element_rect()` is deprecated as of ggplot2 3.4.0.
## ℹ Please use the `linewidth` argument instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.

Gamma estimation

Gof: Our’s method ( effect)

Gof: K’s method (homo effect)